Finsler streamline tracking with single tensor orientationdistribution function for high angular resolution diffusionimagingCitation for published version (APA):Astola, L. J., Jalba, A. C., Balmashnova, E., & Florack, L. M. J. (2011). Finsler streamline tracking with singletensor orientation distribution function for high angular resolution diffusion imaging. Journal of MathematicalImaging and Vision, 41(3), 170-181. https://doi.org/10.1007/s10851-011-0264-4

DOI:10.1007/s10851-011-0264-4

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J Math Imaging Vis (2011) 41:170–181DOI 10.1007/s10851-011-0264-4

Finsler Streamline Tracking with Single Tensor OrientationDistribution Function for High Angular Resolution DiffusionImaging

Laura Astola · Andrei Jalba · Evgeniya Balmashnova ·Luc Florack

Published online: 11 February 2011© The Author(s) 2011. This article is published with open access at Springerlink.com

Abstract We introduce a new framework based onRiemann-Finsler geometry for the analysis of 3D imageswith spherical codomain, more precisely, for which eachvoxel contains a set of directional measurements repre-sented as samples on the unit sphere (antipodal points iden-tified). The application we consider here is in medical imag-ing, notably in High Angular Resolution Diffusion Imaging(HARDI), but the methods are general and can be appliedalso in other contexts, such as material science, et cetera,whenever direction dependent quantities are relevant. Find-ing neural axons in human brain white matter is of signifi-cant importance in understanding human neurophysiology,and the possibility to extract them from a HARDI imagehas a potentially major impact on clinical practice, such asin neuronavigation, deep brain stimulation, et cetera. In thispaper we introduce a novel fiber tracking method which is ageneralization of the streamline tracking used extensively inDiffusion Tensor Imaging (DTI). This method is capable offinding intersecting fibers in voxels with complex diffusion

L. Astola (�)Biometris, Plant Science Group, Wageningen University &Research Centre, Droevendaalsesteeg 1 Building 107, 6708 PBWageningen, The Netherlandse-mail: laura.astola@wur.nl

A. Jalba · E. Balmashnova · L. FlorackDepartment of Mathematics and Computer Science, EindhovenUniversity of Technology, Den Dolech 2, 5612AZ Eindhoven,The Netherlands

A. Jalbae-mail: a.c.jalba@tue.nl

E. Balmashnovae-mail: e.balmachnova@tue.nl

L. Floracke-mail: l.m.j.florack@tue.nl

profiles, and does not involve solving extrema of these pro-files. We also introduce a single tensor representation for theorientation distribution function (ODF) to model the proba-bility that a vector corresponds to a tangent of a fiber. Thesingle tensor representation is chosen because it allows anatural choice of Finsler norm as well as regularization viathe Laplace-Beltrami operator. In addition we define a newconnectivity measure for HARDI-curves to filter the mostprominent fiber candidates. We show some very promisingresults on both synthetic and real data.

Keywords ODF · DTI · HARDI · Riemann geometry ·Finsler geometry · Fiber tracking

1 Introduction

Diffusion Tensor Imaging (DTI) is a non-invasive magneticresonance imaging modality to measure molecular motionof water in biological tissue. It is primarily used to im-age muscular tissue and tissue consisting of densely packedneural axons such as brain white matter. A typical voxel sizeis 1 mm3, while a typical diameter of a single neural axonis in the order of micrometers. An important challenge isto extract the axonal architecture and to find the fiber bun-dles (tractography) connecting different regions of the brain(connectivity). It is assumed that the local diffusion profileis indicative of the gross fiber orientation distribution in avoxel, since there is more diffusion along the direction ofelongation of axons than across these structures.

The DTI model yields an ellipsoidal profile as the levelset of the water diffusivity profile after finite-time diffusion.Such a profile predicts the fiber directions well only in casethe voxel contains fibers with a single preferred direction. Asa symmetric and positive definite (SPD) tensor, the diffusion

J Math Imaging Vis (2011) 41:170–181 171

tensor (or its inverse) can be interpreted as a Riemann met-ric tensor. For this reason several authors have applied toolsfrom Riemannian geometry to the analysis of DTI data [4,6, 17, 21, 30].

High Angular Resolution Diffusion Imaging (HARDI)-is a collective name for techniques that capture more di-rectional measurements, improving the angular resolution.HARDI requires a model of local diffusion that is morecomplex than the rank-two DTI tensor model [24, 35, 39].Finsler geometry can incorporate more general local normfunctions than one induced by a Riemannian inner productand is indeed applicable to the analysis of HARDI data [5,27].

From the various HARDI models we adopt here the so-called Q-ball imaging technique [3, 15, 36] and modelHARDI measurements using spherical tensors, i.e. homo-geneous polynomials restricted to the sphere. However, ourmethod is by no means restricted to Q-ball imaging. Re-cently more advanced methods to reconstruct the ODF orthe so-called orientation probability density function haveemerged [2, 34] and our method is applicable to these aswell.

Using a tensor representation, one can construct a Finslernorm suitable for the analysis of diffusion profiles that canhave a more general shape than that of an ellipsoid. On theother hand, to regularize noisy HARDI data based on heatdiffusion on the sphere, we need a decomposition of the ten-sor into homogeneous and harmonic components that are theeigenfunctions of the Laplace-Beltrami operator [18, 29].We show how to do this in one step without iteratively fittingdifferent order harmonics. We also introduce a novel methodfor fiber tracking in HARDI data based on Finsler geometry.The idea of Finsler geometry in a HARDI setting has beenproposed by Melonakos et al. [27], who adopted a dynamicprogramming approach to compute optimal curves with re-spect to a Finsler norm, without computing Finsler metrictensors and associated eigensystems as we do below. In con-trast to their work, we use a tensor model for which the met-ric can be computed a priori, ODE-based tracking, differenthomogenization technique, and explicit Tikhonov regular-ization based on Laplace-Beltrami diffusion. Regularizationis critical, since HARDI data tend to have low SNR.

This paper is organized as follows: In Sect. 2 we showhow to compute the regularized orientation distributionfunction (ODF) in a tensor framework. In Sect. 3 we com-pute Finslerian diffusion tensors from this ODF. In Sect. 4we extend DTI-streamline tracking to Finslerian streamlinetracking for HARDI data, to allow crossings of fibers. Fi-nally in Sect. 5 we show some promising experimental re-sults on both simulated and real HARDI data.

2 Orientation Distribution Function on a SphericalTensor Field

In HARDI literature spherical harmonics are a natural andpopular choice to represent functions on the sphere [15, 23].Nevertheless, in this paper we prefer an equivalent tensor—isomorphic to homogeneous polynomial—representation[18, 38], since such a description is more natural in a dif-ferential geometric approach.

A (covariant) tensor D of degree n at a point p on a dif-ferentiable manifold M can be seen as a multilinear mapping

D(p) : TpM × · · · × TpM︸ ︷︷ ︸

n times

→ R (1)

in which TpM denotes the tangent space at p ∈ M [26]. Thepoint p has coordinates x = (x1, x2, x3) ∈ R

3 relative to alocal coordinate chart, but explicit reference to this pointwill often be suppressed for ease of notation. All direction-ally dependent functions used below depend on this positionx, and on a local direction vector y ∈ R

3. The vector y mayor may not be confined to the unit sphere (it should be clearfrom the context if a constraint is applicable). In the lattercase it may be expressed in spherical coordinates θ and ϕ,but it is tacitly understood that in this case antipodal pointsare to be identified (implying that the functions under con-sideration are symmetrical).

For example, a second order symmetric spherical tensorin R

3 can be written as

Dijyiyj = D11y

1y1 + D12y1y2 + D13y

1y3

+ D21y2y1 + D22y

2y2 + D23y2y3

+ D31y3y1 + D32y

3y2 + D33y3y3

= D11y1y1 + 2D12y

1y2 + 2D13y1y3

+ D22y2y2 + 2D23y

2y3 + D33y3y3 (2)

suppressing the fiducial point p in the notation and defining

y = (y1, y2, y3)

= (sin θ cosϕ, sin θ sinϕ, cos θ), (3)

with the following summation convention

aibi :=

∑

i

aibi . (4)

To justify the choice we recall that spherical harmonics arein fact homogeneous harmonic polynomials restricted tounit sphere [29]. On the other hand, the choice of using ten-sors instead of equivalent homogeneous polynomials makessome operations elementary and intuitive. This is especially

172 J Math Imaging Vis (2011) 41:170–181

true for our case where only even order symmetric tensorsare involved. A symmetric tensor here means that

Di1···in = Dσ(i1···in), (5)

for any index permutation σ .In Q-ball imaging the diffusion profile Ψ is obtained as

Ψ (y) =∫ ∞

0P(ry)dr, (6)

where P(ry) is the ensemble-average probability that a par-ticle is displaced from a fiducial initial point x0 (implicit inthe notation) to x0 + ry. In [11] it is shown that assumingshort diffusion gradient pulses in the scanning protocol, therelation between signal S(q) and P(r) is

P(r) =∫

R3S(q)ei2πr·qdq, (7)

where q is the wave vector i.e. a unit vector encoding thedirection and pulse duration. Using this relation, in [35] it isfurther shown that Ψ in (6) can be estimated by the Funk-Radon transform of the Fourier transform of P(r). In [15]this result is applied to the case that the signal is modeledwith spherical harmonics and it is shown that the ODF on asingle shell can be approximated as

Ψ (y)|r=1 ≈ 2π

n∑

�=0

�∑

m=−�

c�mY �m(y)P�(0), (8)

where P� denotes the Legendre polynomial of degree �. Inparticular we have

P�(0) =⎧

⎨

⎩

0 � odd

(−1)�/2 1·3·5···(�−1)2·4·6···� � even.

(9)

The main benefit of this approach is that one can computethe ODF in a fast and robust way using spherical harmonics(SH), and also apply Laplace-Beltrami regularization (to anyorder), which is a generalization of Gaussian smoothing forL2 functions on R

2 to L2 functions on the sphere S2 [10].

We use this result, but replace the spherical harmonics bya tensor, cf. [38], which is more practical for applicationsusing Finsler geometry.

We remark that recently a more accurate approach forODF reconstruction has been given by Aganj et al. in [2].However the tensor approach in this paper does not dependon the method with which the ODF is built. As long as thereis an analytical expression for the ODF, we can constructregularized Finsler norms and metrics essential for the Fins-lerian fiber tracking.

For comparison we recall the procedure of fitting spher-ical harmonics to HARDI data according to [15] and ap-plying Laplace-Beltrami diffusion on the harmonic compo-nents [16].

This is done as follows.

1. A set of real bases {Yi}Ni=1 of spherical harmonics up tothe desired order is fitted to data sampled on the sphere{S(θk,ϕk) | k = 1, . . . ,m} using least squares:

S(θ,ϕ) =N

∑

i=1

ciYi(θ,ϕ). (10)

Here the index i collectively denotes the indexed pair(�,m) as used in (8), partially ordered along incremental�-values. The order � for a given index i will henceforthbe denoted by �i .

2. The signal is regularized using Laplace-Beltrami diffu-sion [16, 18, 19]:

S(θ,ϕ, τ ) =N

∑

i=1

e−τ�i (�i+1)ciYi(θ,ϕ). (11)

3. Each spherical harmonic Yi of order �i is multiplied bythe factor 2πP�i

(0).4. Then

ODF(θ,ϕ, τ ) :=N

∑

i=1

2πP�i(0)e−τ�i (�i+1)ciYi(θ,ϕ).

(12)

Returning to our tensor approach, the principle is thesame as in the recipe above, but in addition this approachallows us to construct Finsler norms related to the local dif-fusivity. Moreover, it does not require reference to or storageof spherical harmonics. In this approach the previous stepsare replaced by the following:

1. We fit a nth order tensor D (recall that n is always even)to the sampled data on the sphere:

S(y) = D(y) = Di1···inyi1 · · ·yin, (13)

using the linear least squares. For details of the fittingprocedure one may consult [14]. This is equivalent to fit-ting a polynomial consisting of nth order monomials tothe data in such a way that the error at measured pointsis minimal in L2(S

2) norm. To ensure that fourth-ordertensor is always positive definite we can make use of amethod suggested by Barmpoutis et al. in [9] and im-proved in [20]. For higher order tensors we may applythe PSDT (positive semidefinite diffusion tensor) modelof Qi et al. in [32].

2. From the tensor D we compute the harmonic compo-nents Hn,Hn−2, . . . ,H0, for which

�k/2Hk(y) = 0, k = n,n − 2, . . . ,0, (14)

J Math Imaging Vis (2011) 41:170–181 173

where � is the standard Laplace operator, and decom-pose D as follows:

D(y) =n

∑

k=0

Hk(y)((y1)2 + (y2)2 + (y3)2)(n−k), (15)

where when evaluating this function, we can simplify thelast element in each summand since

((y1)2 + (y2)2 + (y3)2)(n−k) = 1, (16)

for y is a unit vector. The harmonic components Hk(y)

are in fact eigenfunctions of the Laplace-Beltrami oper-ator on the sphere, with eigenvalues k(k + 1) [29]. Thismeans that a harmonic component Hk is actually a lin-ear combination of spherical harmonics Y k

m and can beregularized in the same manner.

This allows us to use the results for Q-ball imag-ing [15, 36]. The harmonic components can be easilycomputed using the so-called Clebsch-projection [29].We have included a simple example on how to computeharmonic components of a second order polynomial inthe Appendix. The extension to higher even order poly-nomials is straightforward.

3. Laplace-Beltrami regularization can be applied to eachHk(y):

D(y, τ) =n

∑

k=0

e−k(k+1)τHk(y). (17)

4. Similar to (12) we now have

ODF(y, τ ) :=n

∑

k=0

2πP�k(0)e−k(k+1)τHk(y). (18)

Thus even the knowledge of a basis of real spherical har-monics is not needed. We remark that an nth order symmet-ric tensor in dimension three has

No =(

n + 2

n

)

= (n + 2)(n + 1)

2(19)

independent terms. Thus an nth order tensor is determinedby No numbers. The Clebsch decomposition can be com-puted very fast, since all the coefficients in harmonic compo-nents are linear combinations of the No terms in the originaltensor. This means that we can compute the components ofthe ODF-tensor with a regularization parameter τ , by simplymultiplying the components of the signal-tensor by a matrix.We give an explicit example of this in the Appendix.

Now that we have a tensor representation of the local dif-fusion probability, we use it to construct generalized, Finslerdiffusion tensors. From these one can compute principaleigenvectors, since at each point and to every tangent vectorthere is a corresponding second order tensor approximatingthe local diffusivity, as we shall see in the following section.

Fig. 1 (Color online) Top: Afourth order tensor fitted to asample of measurements (reddots). Bottom: For a givendirection vector (here green andblue) we have an ellipsoid withcorresponding color thatapproximates the diffusionprofile

3 From Orientation Distribution Function to Metric

Riemannian geometry extends analytical geometry to curvedspaces and equips these with a local inner product, whichin turn allows computations of lengths and angles. Rie-mannian distances thus generalize the theorem of Pythago-ras into skew coordinates. A norm function induced by aRiemannian inner product is a special case of a Finsler normfunction [8, 12, 33]. A Finsler norm depends on both posi-tion and direction. A general convex norm can capture morecomplex angular dependence than the Riemann metric ten-sor. For an illustration, see Fig. 1.

In [5] it is shown that a Finsler structure can be de-fined on a HARDI image and the details of computingthe metric/diffusion tensor from HARDI-ODF function aredescribed. To make this paper self-contained, we repeathere the algorithm to compute generalized diffusion tensorsg(x, y) from the raw HARDI signal S(x, y):

1. The starting point is a single nth order tensor F (x, y)

representing the ODF. We obtain a homogeneous func-tion F(x, y) as follows1

F(x, y) = F (x, y)1/n = (

Fi1···in (x)yi1 · · ·yin)1/n

. (20)

2. When F(x, ·) is strongly convex (true with typicalHARDI data), we obtain a symmetric positive definitetensor:

gij (x, y) = 1

2

∂F 2(x, y)

∂yi∂yj. (21)

1Here homogeneity means F(x,λy) = λF(x, y). Recall also that

F (x, y) is non-negative for all x, y.

174 J Math Imaging Vis (2011) 41:170–181

In case the norm F is a square root of an inner product (Rie-mann metric), then g = F .

We have here two distinct heuristic interpretations forour applications. When applying Finsler geometry, we dealwith norms and inner products that define distances. Thenit is intuitive to replace ODF with the multiplicative inverseof ODF in all computations, for larger diffusivity impliesshorter travel time from the diffusing particle point of view.Since water molecules in human tissue have constant aver-age velocity v, the direct proportionality of (tissue structureinduced) distance and diffusion time can be deduced triv-ially from �x = v�t .

On the other hand, when we want to work directly withthe probabilistic diffusion profile and generalize the stream-line tracking to HARDI case, we adopt directly the conven-tion F = ODF, and compute the direction dependent dif-fusion tensor as in (21). This diffusion tensor describes aquadratic form corresponding to small perturbations of y,which can be seen also from the following second order Tay-lor expansion of F 2 with fixed x (suppressed in the notationhenceforth). Consider the one-dimensional case for simplic-ity, and denote F ′ = d

dyF , then

F 2(y + δ)

= F 2(y) + (F 2(y))′δ + 1

2

(

F 2(y))′′

δ2 + O(δ3)

≈ F 2(y) + 2F(y)F ′(y)δ + 1

2

(

F 2(y))′′

δ2. (22)

Using also the fact that

F(y + δ) = F(y) + F ′(y)δ + O(δ2), (23)

we get

1

2

(

F 2(y))′′ ≈ (F (y + δ) − F(δ))2

δ2. (24)

4 Finslerian Streamlines

4.1 Background

The most popular way to track axon(bundle)-like curves inDTI data is to follow the principal eigenvectors of diffu-sion tensors Dij (x). This approach requires that the curvec : [0,1] → R

3 satisfies the following equations

⎧

⎪⎨

⎪⎩

c(t) = arg max|h|=1{Dij (c(t))hihj },c(0) = p,

c(0) = V.

(25)

The solution to this equation is a geodesic from a point p

to a sphere S2(p, ε) for some small ε, but not necessarily a

Fig. 2 Left: A second ordertessellation gives theequiangular unit vectors usedhere as initial directions for fibertracking at the initial point/voxel

geodesic between c(0) and c(1). This tracking scheme canbe extended to HARDI framework using Finsler geometryby simply replacing Dij (c(t)) with Dij (c(t), c(t)), where

c(t) = limδ→0

c(t) − c(t − δ)

δ. (26)

4.2 Algorithm

Our tracking algorithm belongs to the class of the so-calledfiber assignment by continuous tracking (FACT) algorithms[28]. The practical implementation is similar to that of thestreamline tracking in DTI setting. One difference is thatsince the diffusion profile is not ellipsoidal, but (potentiallymuch) more complex, so that there is no unique initial prin-cipal eigenvector at the initial point of tracking. Instead westart tracking in all gradient directions. The user can definethe gradients to e.g. correspond to the points of a tessella-tion of any order of choice. Here we have used the secondorder tessellation (zeroth order being the icosahedron) with54 gradients (see Fig. 2).

Another difference is that there is no unique second or-der local diffusion tensor per point. Instead there is a uniquesecond order local diffusion tensor corresponding given thedirection of arrival (26) at a given point. This local diffu-sion tensor is computed as in (21) in Sect. 3. The formulafor the appropriate local diffusion tensor is easily computedbeforehand. For example, if we have a fourth order tensor,

D4(x, y) := Dijkl(x)yiyj ykyl (27)

describing the diffusion profile, the Finsler bi-linear form(21) corresponding to the homogenized form

F(x, y) := (D4(x, y))1/4 (28)

is

gij (x, y)

= 1

2

∂2F 2(x, y)

∂yi∂yj

= 1

2

∂2D1/24

∂yi∂yj

J Math Imaging Vis (2011) 41:170–181 175

Table 1 Comparison of the algorithmic structure between the pro-posed Finsler streamline tracking, DTI-streamline tracking and amethod that follows the maxima of the ODF

DTI/Riemann HARDI/maxima HARDI/Finsler

find maxima of compute Hessian

ODF of ODF

find eigensystem find eigensystem generalized

diffusion tensor diffusion tensor

follow principal follow maxima follow principal

eigenvector eigenvector

= 3D−1/24 Dijqp(x)yqyp

− 2D−3/24 Dipkl(x)ypykylDqjmn(x)yqymyn, (29)

where we have put D4 := D4(x, y) for brevity.Now we have a bi-linear form at every position so that if

we supply the vector y of the direction of arrival, it returnsa local diffusion tensor.

If the direction of arrival is close to the principal eigen-vector of this local diffusion tensor, we take a step forward,if not we stop. The familiar anisotropy criterion for DTI,which stops tracking if the diffusion tensor is isotropic, canbe applied to this local diffusion tensor as well.

Thus in a nutshell the algorithm is as follows: If the tan-gent of the streamline is in reasonable alignment with theprincipal eigenvector defined by the local diffusion tensor(corresponding to the tangent) and the fractional anisotropyof the local diffusion tensor is high enough, proceed, oth-erwise stop. A comparison between the essential steps inFinsler streamline tracking, DTI-streamline tracking and amaxima extraction method is summarized in Table 1. Thecomputation of a diffusion tensor or ODF (in terms of spher-ical harmonics or equivalently in terms of a monomial ten-sor) is considered as a preprocessing step and is not includedin the table.

The only difference between our method and the stan-dard DTI streamline tracking is the extra step in which theHessian of the ODF is computed. However, in Sect. 5.3 weshow that this additional step is relatively inexpensive andcan be performed with less overhead. On the other hand,standard ODF-based tracking methods [13] typically requirefinding multiple, local maxima of the ODF. This in turn canbe expensive in terms of computations [1]. For example,one can attempt to find all the local maxima of the ODF bynumerical, iterative approaches, e.g. Newton method. How-ever, such an approach would also require an additional ex-haustive, two-dimensional search, to guarantee that all max-ima are found, with accuracy within the discretization res-olution. Thus, the complexity of such an attempt would bequadratic in the desired accuracy. Although, it is possible toreduce the search space to one dimension [1], the computa-tion of all local maxima of the ODF remains an expensive

operation. Moreover, if the streamline tracking algorithm isrequired to follow all local maxima, branching (or splitting)of the streamlines has to be allowed [13], which further in-creases the computational requirements. Finally, the simplestopping criterion based on the computation of the fractionalanisotropy (FA), now relies on the more computationally in-volved, generalized anisotropy measure [35, 40].

In Sect. 5.3 we show that even if only one ODF (global)maximum is used for advancing the streamline, the result-ing method is still more expensive than ours. Note that, tofind the global maximum we simply evaluate the ODF ina number of search directions (i.e. scanning directions) anddeem the largest value as the sought maximum. Clearly sucha naive approach can potentially be very inaccurate, yet itis certainly faster than both the iterative approach and themethod in [1].

4.3 Connectivity measure

The connectivity measure for an arbitrary curve defined hereindicates the relative amount of diffusivity along this curve.Higher value implies greater diffusivity. In isotropic regionsits value is one. In DTI setting, to compute the so-called con-nectivity [7, 31] of a curve (fiber-candidate), we comparethe Euclidean length of the curve to its Riemannian length(determined by the diffusion tensors along the curve) anddefine a connectivity measure m(γ ) of a curve γ (t) as fol-lows:

m(γ ) =∫

√

ηij (γ (t))γ (t)i γ (t)j dt∫ √

gkl(γ (t))γ (t)kγ (t)ldt, (30)

where the ηij (γ ) represents the covariant Euclidean metrictensor, which in Cartesian coordinates reduces to the con-stant identity matrix, γ the tangent to the curve γ and gkl(γ )

the Riemann-metric tensor.We can generalize this connectivity measure directly to

HARDI setting.

m(γ ) =∫

√

ηij (γ (t))γ (t)i γ (t)j dt∫ √

gkl(γ (t), γ (t))γ (t)kγ (t)ldt, (31)

where gkl(γ, γ ) is the Finsler-metric tensor (which dependsnot only on the position but also on the direction of thecurve). Since gij (x, y)yiyj = F 2(x, y), this can be simpli-fied to

m(γ ) =∫

√

ηij (γ (t))γ (t)i γ (t)j dt∫

F(γ (t), γ (t))dt. (32)

This measure can be computed along any curve. In Sect. 5we show results of fiber tracking and connectivity measures.

176 J Math Imaging Vis (2011) 41:170–181

Fig. 3 (Color online) Left:Streamlines in a noise freetensor field simulating a 65°crossing of fiber bundles.Streamlines are coloredaccording to their connectivity.High (low) connectivity is colorcoded as red (blue). Right: Samefield with Rician noise added

Fig. 4 Left: Streamlines in anoise free tensor fieldsimulating a 30° crossing offiber bundles. Note that smallerangle results in kissingstreamlines. Right: Same fieldwith Rician noise added

Fig. 5 Second order tensorfield on real data andDTI-streamlines with black seedpoints. In DTI, intersectingfibers are impossible since thereis only one tangent vectoroccupying a point. Apparentcrossings are due to theprojection of spatial curves toimage plane

5 Experimental Results

5.1 Simulated Data

We computed Finsler streamlines in tensor fields that sim-ulate two fiber bundles crossing at angles of 30 and 65 de-grees. The tensors in the crossing area are generated usingthe Gaussian mixture model [37] assuming signal with b-value 1000 s/m2. Noisy images were generated by simulat-ing Rician noise with SNR around 15, in our case this corre-sponds to the inverse of standard deviation of the (Gaussian)noise added to the real and complex parts of the signal.We solved the Finsler streamlines (25) using a third orderRunge-Kutta model for integration [22]. If the fractionalanisotropy (FA) became less than 0.2 or if the inner productof the tangent of the curve and the local principal eigenvec-tor became less than 0.1, the tracking stops. These parame-ters are similar to those used in the standard DTI-fiber track-ing. We have colored the fibers using a temperature map ac-cording to their connectivity (32) so that fibers with high-est (lowest) connectivity are colored red (blue). See Figs. 4and 3.

5.2 Real Data

We used a HARDI scan of a human brain with b-value 1000s/m2, and 132 gradient directions. As initial points we used

30 voxels in the area where the major fiber bundles calledCorpus Callosum and the Corona Radiata are known to crosseach other. The parameters used are identical to those usedin simulated data except that the critical value of FA wasset to 0.09. For comparison we added also the correspond-ing second order tensor field and a result of standard DTI-tracking in Fig. 5. Finsler streamlines in Fig. 6 indeed doshow crossings in those regions where this is to be expected,unlike standard DTI-tracking. In Fig. 7 we used otherwiseidentical settings, but increased the number of initial points.

5.3 Performance Comparison

In this section we compare the performance of our method(Finsler) with standard DTI streamline tracking (DTI) anda simple method based on computing the global maximumof the ODF (ODF-gmax), as discussed in Sect. 4.2. In thefirst experiment we measured the time required by each ofthese methods to track a streamline of (approximately) equallength. The number of search directions of the (global) max-imum is set equal to the number of scanning directions,here 54. The same number of directions is also used byour Finsler tracking method at the initial point/voxel, seeFig. 2. Table 2 shows the performance of each method. As itcan be seen, our method performs similarly to standard DTItracking and is around three times faster than the ODF-gmaxmethod.

J Math Imaging Vis (2011) 41:170–181 177

Fig. 6 Fourth order tensorglyphs on real HARDI data andFinsler streamlines with seedpoints. Although thesestreamlines are generally not onthe same plane, they do actuallycross each other in a way that isin accordance with theanatomical knowledge of humanbrain [25]

Fig. 7 (Color online) A resultof Finsler fiber tracking, whenusing a larger set of initialpoints. Besides the crossings ofCorona Radiata and CorpusCallosum, also some Cingulumfibers are clearly visible. Thecolor of the fibers indicate themajor orientation of the fiber.Blue: top-bottom, red: left-right,green: back-front

Table 2 Performance of the three methods when tracking a streamlineof (approximately) equal length. Third column gives the timing (mil-liseconds) for each method. The number of search and initial directionsis 54

Method Euclidean length Timing

DTI 27.3 1.5

Finsler 27.5 1.9

ODF-g max 27.1 6.8

In the next experiment we increased the number of di-rections (search for ODF-gmax and initial for Finsler) to250, see Table 3. Note that in this case our method is about8 times faster than the ODF-gmax method. The large dropin performance of the ODF-gmax method can be partly ex-plained by the fact that now five times more evaluations arerequired for maxima computation. Moreover, evaluating thegeneralized FA becomes also more expensive.

Finally, tracking streamlines using the data set and initial-ization from Fig. 6 took 0.08 seconds by the DTI method,0.2 seconds with our method and 2.8 seconds using ODF-

Table 3 Performance of Finsler and ODF-gmax methods when track-ing a streamline of (approximately) equal length. Third column givesthe timing (milliseconds) for each method. The number of search andinitial directions is 250

Method Euclidean length Timing

Finsler 27.1 2.9

ODF-g max 27.2 23.6

gmax. The number of search/initial directions in this exper-iment was 54.

6 Summary and Conclusions

We have presented a method with which one can obtain aregularized orientation distribution function from the ten-sor components (polynomial coefficients) representing theHARDI signal, with a single matrix multiplication. Themethod includes a parameter for Laplace-Beltrami regular-ization, which solves the heat equation on the sphere, and is

178 J Math Imaging Vis (2011) 41:170–181

therefore suitable for noisy spherical data such as HARDI.The choice of the order of the tensor approximating the sig-nal has the effect of an additional regularization. This trun-cation however in practice cannot be avoided and is ulti-mately bounded by the number of measurements.

We have also shown a novel method for fiber tracking inHARDI case, which is a true generalization of the streamlinetracking in the diffusion tensor setting. Our method is basedon Finsler geometry and does not require solving for localmaxima of the tensors, which typically is a requirement forother methods.

In addition we have introduced a measure for filteringout those streamlines that are not well connected, i.e. thathave small overall diffusivity. Our connectivity measure canbe applied to any curve in a regular space equipped with anorm.

Acknowledgements The Netherlands Organization for ScientificResearch is gratefully acknowledged for financial support. We thankPaulo Rodrigues and Vesna Prckovska at the Biomedical Image Analy-sis group of the department of Biomedical Engineering at EindhovenUniversity of Technology for kindly providing the real and simulatedHARDI data.

Open Access This article is distributed under the terms of the Cre-ative Commons Attribution Noncommercial License which permitsany noncommercial use, distribution, and reproduction in any medium,provided the original author(s) and source are credited.

Appendix

Here we show an example of using the formula of Cleb-sch to project a polynomial of degree n to a harmonic and anon-harmonic part. Applying the projection iteratively oneobtains a decomposition of the original polynomial to a sumof harmonic polynomials of degrees up to n [29]. Let Dn(y)

be a homogeneous polynomial of degree n. In our case thisis

Dn(y) = Di1···inyi1 · · ·yin, (33)

where

y = (sin θ cosϕ, sin θ sinϕ, cos θ), (34)

i.e. a homogeneous polynomial restricted to the sphere inR

3. The Clebsch projection operator in dimension three isas follows

Pn :=[ n

2 ]∑

k=1

(−1)k−1

4k

�(n − k + 12 )

k!�(n + 12 )

|y|2k�k, (35)

where � refers to the Gamma function:

�(z) =∫ ∞

0tz−1e−t dt. (36)

Computing

Hn(y) = Dn(y) − Pn(Dn)(y), (37)

gives Hn(y) which is a homogeneous and harmonic poly-nomial of degree n. For proof see [29]. Next we show aconcrete example of computing Laplace-Beltrami regular-ized ODF by simply multiplying the nth order tensor com-ponents fitted to the signal by a matrix. For simplicity weshow this in degree two, but it extends to all even degrees.Let

S = (S11, S12, S13, S22, S23, S33) (38)

be the six components of the symmetric tensor approximat-ing the DT-MRI signal. Then the local second order polyno-mial describing the signal is

D2(y) = S11y1y1 + 2S12y

1y2 + S13y1y3

+ S22y2y2 + 2S23y

2y3 + S33y3y3, (39)

where y is as in (34). Clebsch projection (35) applied to apolynomial of degree two (39) gives

P(D2(y)) = 1

6|y|2�D2(y)

= 1

3

(

(y1)2+(y2)2+(y3)2) (S11+S22+S33)

= 1

3(S11 + S22 + S33)

= H0(y), (40)

which is the zeroth order solid harmonic. The second ordersolid harmonic is then the remainder

H2(y) = D2(y) − P(D2(y))

=(

2

3S11 − 1

3S22 − 1

3S33

)

(y1)2

+(

2

3S22 − 1

3S11 − 1

3S33

)

(y2)2

+(

2

3S33 − 1

3S22 − 1

3S11

)

(y3)2

+ 2S12y1y2 + 2S13y

1y3 + 2S23y2y3. (41)

Multiplying these harmonic components H0,H2 with ap-propriate terms in (9), i.e. 2πP0(0) = 2π and 2πP2(0) =−π , we obtain the ODF:

D2(y) = π(2H0(y) − H2(y)). (42)

To apply Laplace-Beltrami smoothing as in (17), we mul-tiply each harmonic component with the order dependent

J Math Imaging Vis (2011) 41:170–181 179

regularization term as follows

D2(y, τ ) = π(

e−τ(0+1)02H0(y) − e−τ(2+1)2H2(y))

. (43)

Thus the zeroth order part of the regularized ODF is

H0(τ ) = 2π

3(S11 + S22 + S33), (44)

which can be expanded to second order homogeneous poly-nomial by taking the Kronecker product of this scalar andidentity matrix I3 (similarly for higher order polynomials[5]). By doing this we obtain the following representationfor H0(τ ):

H0(y, τ ) = 2π

3(S11 + S22 + S33)((y

1)2 + (y2)2 + (y3)2).

(45)

Recall that ((y1)2 + (y2)2 + (y3)2) = 1.The regularized second order part of the ODF is:

H2(y, τ ) = −πe−6τ

[(

2

3S11 − 1

3S22 − 1

3S33

)

(y1)2

+(

2

3S22 − 1

3S11 − 1

3S33

)

(y2)2

+(

2

3S33 − 1

3S22 − 1

3S11

)

(y3)2

+ 2S12y1y2 + 2S12y

1y3 + 2S23y2y3

]

. (46)

Now, the sum H0(y, τ ) + H2(y, τ ) gives exactly the regu-larized second order ODF. With some reverse engineering,we can easily determine the transformation matrix that takessignal S(y) to regularized ODF: H0(y, τ ) + H2(y, τ ).

Given a symmetric second order tensor S(y) (38) rep-resenting the raw HARDI signal, the j th term in the com-ponents vector (cf. (38)) of second order tensor SODF(y, τ )

representing τ -regularized ODF is then

SODF(y, τ )j =6

∑

k=1

2πcjkS(y)k, (47)

where

c11 = 1

3(1 − e−6τ ),

c12 = c21 = 0,

c13 = c31 = 0,

c14 = c41 = 1

6e−6τ ,

c15 = c51 = 0,

c16 = 1

6e−6τ ,

c22 = 2,

c23 = c32 = 0,

c24 = c42 = 0,

c25 = c52 = 0,

c26 = c62 = 0,

c33 = 2,

c34 = c43 = 0,

c35 = c53 = 0,

c36 = c63 = 0,

c44 = 1

3(1 − e−6τ ),

c45 = c54 = 0,

c46 = 1

6e−6τ ,

c55 = 2,

c56 = c65 = 0,

c66 = 1

3(1 − e−6τ ).

This matrix as well as the corresponding higher ordertensors have to be computed only once. Thus to model aHARDI ODF with a polynomial of order n with regulariza-tion parameter τ , one needs only to fit an nth order polyno-mial to the raw signal, multiply it with a precomputed matrixcij and assign a parameter τ of choice.

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Laura Astola received her M.Sc.degree in mathematics at HelsinkiUniversity, Finland, in 2000. In2009, she obtained her degree ofLicentiate of Engineering fromHelsinki University of Technology.She received her Ph.D. degree witha thesis on differential geometryand applications in medical imageanalysis from Eindhoven Universityof Technology in 2010. She is cur-rently a postdoctoral fellow at Bio-metrics in Wageningen Universityand Research Center and Nether-lands Consortium for Systems Biol-

ogy. Her research concerns reconstruction and inference of metabolicand genetic networks using statistical analysis and techniques frompattern recognition and graph theory.

J Math Imaging Vis (2011) 41:170–181 181

Andrei Jalba received the B.Sc.(1998) and M.Sc. (1999) degrees inapplied electronics and informationengineering from the “Politehnica”University of Bucharest, Romania.He holds a Ph.D. in mathematicsand natural sciences (2004) fromthe University of Groningen, TheNetherlands. Currently he is an as-sistant professor in Visualization atthe Eindhoven University of Tech-nology. His research interests in-clude scientific and parallel com-puting with applications in (med-ical) image processing and com-puter graphics.

Evgeniya Balmashnova receivedher M.Sc. degree in mathematics atNovosibirsk State University, Rus-sia, in 2000. In 2001–2003 she fol-lowed postgraduate program Math-ematics for Industry at the StanAckermans Institute in Eindhoven.She received her Ph.D. degree in2007 from Eindhoven University ofTechnology. She is currently a post-doctoral fellow at the Department ofMathematics and Computer Scienceat the same university. Her researchis focused on medical image analy-sis, high angular resolution diffu-

sion imaging and diffusion tensor imaging in particular, multiscaleapproaches in image analysis.

Luc Florack received his M.Sc. de-gree in theoretical physics in 1989,and his Ph.D. degree cum laudein 1993 with a thesis on imagestructure under the supervision ofprofessor Max Viergever and pro-fessor Jan Koenderink, both fromUtrecht University, The Nether-lands. During the period 1994–1995 he was an ERCIM/HCM re-search fellow at INRIA Sophia-Antipolis, France, with professorOlivier Faugeras, and at INESCAveiro, Portugal, with professorAntonio Sousa Pereira. In 1996 he

was an assistant research professor at DIKU, Copenhagen, Denmark,with professor Peter Johansen, on a grant from the Danish ResearchCouncil. In 1997 he returned to Utrecht University, were he becamean assistant research professor at the Department of Mathematics andComputer Science. In 2001 he moved to Eindhoven University of Tech-nology, Department of Biomedical Engineering, were he became anassociate professor in 2002. In 2007 he was appointed full professorat the Department of Mathematics and Computer Science, and estab-lished the chair of Mathematical Image Analysis, but retained a part-time professorship at the former department. His research covers math-ematical models of structural aspects of signals, images, and movies,particularly multiscale and differential geometric representations andtheir applications, with a focus on complex magnetic resonance imagesfor cardiological and neurological applications. In 2010, with supportof the Executive Board of Eindhoven University of Technology, he es-tablished the Imaging Science & Technology research group (IST/e),a cross-divisional collaboration involving several academic groups onimage acquisition, biomedical and mathematical image analysis, visu-alization and visual analytics.